Solve the equation log &#x2061;<!-- ⁡ --> ( z 2 </msup> &#x2212;<!-- − --> 1

deceptie3j

deceptie3j

Answered question

2022-06-27

Solve the equation log ( z 2 1 ) = i π / 2
I set z = x + y i so:
log [ ( x + y i ) 2 1 ] = log ( x 2 + 2 x y i y 2 1 ) = log ( r + i θ ) = i π / 2
than I get x 2 y 2 = 1 and I have no idea how to continue.

Answer & Explanation

kejohananws

kejohananws

Beginner2022-06-28Added 19 answers

It may be easier to note that:
z 2 1 = e i π / 2 = i
So that:
z = ± i + 1
Gybrisysmemiau7

Gybrisysmemiau7

Beginner2022-06-29Added 2 answers

Note that log ( z ) = ln | z | + i arg ( z ). Then you have
log ( z 2 1 ) = ln | z 2 1 | + i arg ( z 2 1 ) = i π 2

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